The question of generalising circle packing to three dimensions was asked in 65677. There is a clear consensus that there is no obvious three dimensional version of circle packing.

However I have seen a comment that circle packing on surfaces and Ricci flow on surfaces are related. The circle packing here is an extension of circle packing to include intersection angles between the circles with a particular choice for these angles. My initial question is to ask for an explanation of this.

My real question should now be apparent. There is an extension of Ricci flow to three dimensions: so is there some version of circle packing in three dimensions which can be interpreted as a combinatorial version of Ricci flow?

3 Answers
3

Actually, there are a number of references by Ben Chow, Feng Luo, and D. Glickenstein on this subject, mostly in two dimensions. Glickenstein's work (Glickenstein was a student of Ben Chow's) is more three-dimensional. Some relevant references are below. The curvature flow approach distinct from the even more popular variational approach (though the two approaches intersect nontrivially).

Not yet mentioned is the interesting definition of Ricci curvature by
Yann Ollivier, a definition especially suited to discrete spaces, such as graphs.
His definition "can be used to define a notion of 'curvature at a given scale' for metric
spaces." For example, he shows how the discrete cube $\{ 0,1 \}^n$
behaves like $\mathbb{S}^n$ in having constant positive curvature,
and possessing an analog of the Lévy "concentration of measure"
(the mass of $\mathbb{S}^n$ is concentrated about its equator).

This last paper by Miller et al. appears to be dimension-agnostic. In particular, a discrete curvature can be defined at any $(n-2)$-cell in a piecewise-flat simplicial manifold simply by considering the area of its dual cell.
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Vidit NandaFeb 8 '13 at 19:17

provides a combinatorial flow for 3-manifolds with boundary consisting of surfaces
with negative Euler characteristic. It deals with the convergence of an initial "piecewise-hyperbolic" metric to an actual hyperbolic metric with geodesic boundary. The analogy with Ricci flow is very very mild, but I hope you may be interested in this reference anyway.